The Binary Linearization Complexity of Pseudo-Boolean Functions

TL;DR

This paper introduces the concept of linearization complexity for pseudo-Boolean functions, analyzing its theoretical properties and demonstrating practical applications in integer linear programming models.

Contribution

It defines linearization complexity, proves its high value for random polynomials, and explores its characterization under different Boolean function restrictions.

Findings

Random polynomials almost surely have high linearization complexity.

Characterizations of linearization complexity depend on Boolean function restrictions.

Integer linear programming models effectively utilize the proposed linearizations.

Abstract

We consider the problem of linearizing a pseudo-Boolean function $f : \{0,1\}^n \to \mathbb{R}$ by means of $k$ Boolean functions. Such a linearization yields an integer linear programming formulation with only $k$ auxiliary variables. This motivates the definition of the linarization complexity of $f$ as the minimum such $k$. Our theoretical contributions are the proof that random polynomials almost surely have a high linearization complexity and characterizations of its value in case we do or do not restrict the set of admissible Boolean functions. The practical relevance is shown by devising and evaluating integer linear programming models of two such linearizations for the low auto-correlation binary sequences problem. Still, many problems around this new concept remain open.